1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points
The lattice of subgroups of a free group
Enric Ventura
Departament de Matemàtica Aplicada III Universitat Politècnica de Catalunya
GAGTA-5 mini-course, 2011
July 5th, 2011. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points
This is work done by different authors during several years, and in different contexts.
We’ll mostly follow a version by Bartholdi-Silva.
Then, we’ll see several applications. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Free monoid
Definition
• A = {a1,..., ar } is a finite alphabet. • A∗ is the free monoid on A. • A language is a subset L ⊆ A∗. ˜ −1 −1 • An involutive alphabet A = {a1,..., ar , a1 ,..., ar }. • Reduced words; reduction ∼;R(A). • Formal word definitions (a−1)−1 = a, (a1 ··· ak )−1 = a−k ··· a−1 . i1 ik ik i1 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Free group
Definition The free group on A, F(A) = A˜∗/ ∼.
Lemma ˜∗ For every w ∈ A , there is a unique u ∈ R(A), s.t. u =F(A) u.
Lemma F(A) is a quotient of A˜∗. The projection is denoted π:
π : A˜∗ → F(A) u 7→ [u] = [ u ].
Definition ˜∗ For a subgroup H 6 F(A), define H = {u | u ∈ H} ⊆ A . 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Automata
Definition Let A be an alphabet. An A-automaton A is an oriented graph with labels from A at the edges, and with a basepoint, A = (V , E, q0), where • V is a finite set (of vertices), • E ⊆ V × A × V is the set of edges,
• q0 ∈ V is the basepoint, such that the underlying undirected graph is connected.
Note that A admits loops, but no parallel edges with the same label.
Definition
An A-automata A = (V , E, q0) is involutive if A is an involutive alphabet and (p, a, q) ∈ E ⇔ (q, a−1, p) ∈ E. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Paths
Form now on, all automata we consider will be involutive.
Definition Let A be an A-automata. • A path γ in A, • the label of a path γ, label(γ) ∈ A˜∗, • reduced path, • notation:p →u q means a path from p to q with label u ∈ A˜∗.
Lemma Let p →u q be a path in A. If u is reduced then p →u q is reduced. The convers is not true. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Trimness
Definition The language of an A-automata A, is
˜∗ u ˜∗ L(A) = {u ∈ A | ∃ q0 → q0} ⊆ A .
Definition An A-automata A is trim if it has no vertices of degree 1 except maybe the basepoint.
Lemma
If A is trim then ∀q 6= q0 there exists a reduced path q0 → q → q0. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Determinism
Definition An A-automata A is deterministic if (p, a, q) ∈ E and (p, a, q0) ∈ E imply q = q0.
Lemma Let A be a deterministic A-automaton. We have, i) if p →u q is reduced then u is reduced, ii) if ∃ p →u q, ∃ p →u q0 then q = q0, iii) if ∃ p →u q, ∃ p0 →u q then p = p0. −1 iv) if ∃ p uvv→ w q, then ∃ p →uw q. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Morphisms
Definition 0 0 0 0 Let A = (V , E, q0) and A = (V , E , q0) be two A-automata. A 0 0 0 morphism A → A is a map ϕ: V → V such that q0ϕ = q0 and (p, a, q) ∈ E ⇒ (pϕ, a, qϕ) ∈ E 0.
Proposition 0 0 0 0 0 Let A = (V , E, q0) and A = (V , E , q0) be two A-automata, A deterministic. Then,
L(A) ⊆ L(A0) ⇔ ∃ morphism ϕ: A → A0.
In this case, ϕ is unique.
This proof will be repeated with variations later. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Morphisms
Corollary If A is deterministic then the only morphism A → A is the identity.
Corollary If A and A0 are deterministic and L(A) = L(A0) then A'A0.
Corollary If A and A0 are deterministic and trim, and L(A)π = L(A0)π, then A'A0. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Flower automaton
GOAL: to construct an algorithmic bijection between the lattice of finitely generated subgroups of F(A), and the set of A-automata deterministic and trim.
Definition Given a finite set of reduced words W ⊆ R(A) ⊆ F(A), we define the flower automaton F(W ) in the natural way.
Observation The flower automaton F(W ) is i) involutive (by construction), ii) trim, iii) deterministic except maybe at the basepoint, iv) L(F(W ))π = hW i 6 F(A). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Stallings folding
We want to make F(W ) deterministic.
Definition Let A be an A-automaton, and suppose e = (p, a, q) and e0 = (p, a, q0) are two different edges (so, q 6= q0). Consider L = {{f , f 0}= 6 {e, e0} | f = (p0, b, q), f 0 = (p0, b, q0) for some p0 ∈ V , b ∈ A˜}. Consider the automata A0 to be A identifying q = q0 (and so, e = e0, 0 0 0 and f = f for every {f , f } ∈ L, if any). We define A A to be a Stallings folding.
The number ` = |L| > 0 is called the lost of the folding. A folding is called critical when it has positive lost. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Stallings folding
Observation 0 0 0 Let A = (V , E, q0) A = (V , E , q0) be a folding with lost ` > 0. Then, |V 0| = |V | − 1 and |E 0| = |E| − 1 − `.
Observation Applying enough foldings to any given A-automata A,
0 k A A ··· A , we obtain a deterministic Ak (in principle, depending on the chosen sequence of foldings). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Stallings folding
Lemma 0 Let A A to be a Stallings folding. Then ˜ a 0 i) If, in A, |{a ∈ A|q → }| > 2 ∀q 6= q0, then the same is true in A . ii) ∃ a morphism ϕ: A → A0 (and so, L(A) ⊆ L(A0)). iii) L(A)π = L(A0)π.
Corollary Let W ⊂ R(A), |W | < ∞, and consider a sequence of foldings F(W ) ··· A to a deterministic A. Then, i) A is deterministic and trim, ii) L(A)π = H = hW i 6 F(A). iii) H ⊆ L(A) ⊆ Hπ−1. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Stallings folding
Lemma 0 Let A A to be a Stallings folding. Then the natural map
0 πA(q0, q0) → πA (q0, q0) γ 7→ γ0 = γϕ (+ canc. f.e.’s),
satisfies i) label(γ)π = label(γ0)π, ii) in the non-critical case, it is injective.
Corollary If all foldings in F(W ) ··· A are non-critical, then the above 0 map, πF(W )(q0, q0) → πA(q0, q0), γ 7→ γ , is injective. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Membership problem
Theorem The membership problem is solvable in F(A): given g, h1,..., hn ∈ F(A), one can decide whether g ∈ H = hh1,..., hni.
(1) can assume W = {h1,..., hn} ⊆ R(A); (2) draw the flower automaton F(W ); (3) apply an arbitrary sequence of foldings until a deterministic automaton F(W ) ··· A; (4) start reading g as (the label of) a path in A, from q0; (5) if not possible theng 6∈ H; (6) if possible (so, in a unique way) but as an open path theng 6∈ H;
(7) if possible as a closed path at q0, theng ∈ H. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Independence of the process
In a sequence of Stallings foldings, F(W ) ··· A, the result will not depend on the process, and even on W , but only on the subgroup H = hW i 6 F(A).
Theorem A depends only on H = hW i, and is called the Schreier graph, Γ(H).
Proposition
Let H 6f .g F(A), choose a finite set of generators W , hW i = H, and let F(W ) ··· A be an arbitrary sequence of Stallings foldings, with A deterministic. Then, \ L(A) = L(B), B where B runs over all possible automata deterministic, trim, and such that H ⊆ L(B) ⊆ A˜∗. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The bijection
Theorem This is a bijection:
{H 6f .g. F(A)} → {A-automata deterministic and trim} H 7→ Γ(H) L(A)π ← A.
Observation Both directions are algorithmic, and fast. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Nielsen-Schreier Theorem
Let A be an A-automata deterministic and trim, and let H = L(A)π 6f .g. F(A). Take a maximal tree T in A and for every e ∈ EA\ ET take
he = label(T [q0, ιe]eT [τe, q0])π ∈ H.
Proposition
{he | e ∈ EA\ ET } is a free basis for H.
Theorem (Nielsen) Every finitely generated subgroup of a free group is free.
Theorem (Schreier) Every subgroup of a free group is free. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Computability of rank and basis
Proposition
There is an algorithm which, given h1 ..., hn ∈ F(A), computes the rank and a basis of H = hh1 ..., hni 6 F(A). More specifically, rg(H) = 1 − |V Γ(H)| + |EΓ(H)| = |W | − `,
where ` is the total lost in the chain F(W ) ··· Γ(H).
Theorem Free groups are hopfian.
Corollary F(A) ' F(B) if and only if |A| = |B|. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The full Schreier graph
Definition An A-automata A is complete if every vertex has an edge going in and an edge going out with each label.
Definition ˜ For H 6f .g. F(A), define Γ(H) to be Γ(H) with infinite trees attached in order to make it complete.
Observation i) Γ(H) is complete ⇔ Γ(˜ H) = Γ(H). ii) ∀ u ∈ A˜∗, ∀ q ∈ V , ∃ q →u in Γ(˜ H). iii) L(Γ(˜ H)) = Hπ−1. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Finite index subgroups
Proposition ˜ Let H 6f .g. F(A) and let T be a maximal tree in Γ(H). Then,
ϕ: V Γ(˜ H) → H\F(A) v 7→ H · lv
is bijective, where lv = label(T [q0, v])π.
Corollary
Let H 6f .g. F(A). Then, H 6f .i. F(A) ⇔ Γ(H) is complete. In this case, [F(A): H] = |V Γ(H)|.
Corollary (Schreier index formula)
Every H 6f .i. F(A) is finitely generated and r(H) − 1 = [F(A): H](|A| − 1). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Finite index subgroups
Corollary
There is an algorithm which, given h1,..., hn ∈ F(A), decides whether H = hh1,..., hni is of finite index in F(A) and, in this case, computes the index and a set of coset representatives.
Corollary
There is an algorithm which, given h1,..., hn, k1,..., km ∈ F(A), decides whether hh1,..., hni = H 6f .i. K = hk1,..., kmi and, in this case, computes the index and a set of coset representatives. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Normality
Corollary
If 1 6= H 6f .g. F(A) is normal then H 6f .i. F(A).
Corollary
There is an algorithm which, given h1,..., hn ∈ F(A), decides whether H = hh1,..., hni is normal in F(A).
Corollary
There is an algorithm which, given h1,..., hn, k1,..., km ∈ F(A), decides whether H = hh1,..., hni is normal in K = hk1,..., kmi. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points M. Hall’s theorem
Definition Let H 6 K 6 F(A). We say that H is a free factor of K , denoted H 6f .f . K , if it is possible to extend a basis of H to a basis of K .
Lemma
For H 6f .g. K 6f .g. F(A), if Γ(H) is a subautomaton of Γ(K ) then H 6f .f . K . The convers is not true.
Theorem (M. Hall)
For every H 6f .g. F(A) there exists K 6f .i. F(A) such that H 6f .f . K 6f .i. F(A). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Residual finiteness
Definition A group G is said to be residually finite if ∀1 6= g ∈ G there exists a finite quotient G/H where 1 6= g ∈ G/H.
Theorem Free groups are residually finite.
Theorem Free groups are residually p, for every prime p. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Howson property
Definition A group G satisfies the Howson property if the intersection of two finitely generated subgroups is again finitely generated.
Theorem Free groups satisfy the Howson property.
Theorem
There is an algorithm which, given h1,..., hn, k1,..., km ∈ F(A), computes a basis of H ∩ K , where H = hh1,..., hni and K = hk1,..., kmi. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Hanna Neumann “Conjecture"
Definition Define the reduced rank of H 6 F(A) as ˜r(H) = max{0, r(H) − 1}.
Theorem
For H, K 6f .g. F(A), ˜r(H ∩ K ) 6 2˜r(H)˜r(K ).
Hanna Neumann “Conjecture"
For H, K 6f .g. F(A), ˜r(H ∩ K ) 6 ˜r(H)˜r(K ).
Theorem (Mineyev, (simpl. Dicks))
For H, K 6f .g. F(A), ˜r(H ∩ K ) 6 ˜r(H)˜r(K ). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Malnormality
Definition g A subgroup of a group H 6 G is malnormal if H ∩ H = 1 for all g 6∈ H.
Proposition
There is an algorithm which, given h1,..., hn ∈ F(A), decides whether H = hh1,..., hni is malnormal in F(A).
Observation u v For H, K 6f .g. F(A), the collection of intersections H ∩ K , moving u, v ∈ F(A), takes only finitely many values, up to conjugacy. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Motivation
In basic linear algebra:
n U 6 V 6 K ⇒ V = U ⊕ L.
In Zn, the analog is almost true:
n 0 0 U 6 V 6 Z ⇒ ∃ U 6fi U 6 V s.t. V = U ⊕ L.
In F(A), the analog is ...
far from true because H 6 K 6⇒ r(H) 6 r(K ) ... 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Motivation
In basic linear algebra:
n U 6 V 6 K ⇒ V = U ⊕ L.
In Zn, the analog is almost true:
n 0 0 U 6 V 6 Z ⇒ ∃ U 6fi U 6 V s.t. V = U ⊕ L.
In F(A), the analog is ...
almost true again, ... in the sense of Takahasi. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Algebraic and transcendental elements
Mimicking field theory...
Definition Let H 6 F(A) and w ∈ F(A). We say that w is algebraic over H if ∃ 1 6= eH (x) ∈ H ∗ hxi such that eH (w) = 1; transcendental over H otherwise.
Observation w is transcendental over H ⇐⇒ hH, wi ' H ∗ hwi ⇐⇒ H is contained in a proper f.f. of hH, wi.
Problem
w1, w2 algebraic over H 6⇒ w1w2 algebraic over H.
H = ha, bab, caci 6 ha, b, ci, and w1 = b, w2 = c. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Algebraic and free extensions
A relative notion works better... Definition Let H 6 K 6 F(A) and w ∈ K . We say that w is K -algebraic over H if ∀ free factorization K = K1 ∗ K2 with H 6 K1, we have w ∈ K1; K -transcendental over H otherwise.
Observation w is algebraic over H if and only if it is hH, wi-algebraic over H.
Observation
If w1 and w2 are K -algebraic over H, then so is w1w2. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Algebraic and free extensions
Definition Let H 6 K 6 F(A). We say thatH 6 K is an algebraic extension, denotedH 6alg K, ⇐⇒ every w ∈ K is K -algebraic over H, ⇐⇒ H is not contained in any proper free factor of K , ⇐⇒ H 6 K1 6 K1 ∗ K2 = K implies K2 = 1.
We say thatH 6 K is a free extension, denotedH 6ff K, ⇐⇒ every w ∈ K is K -transcendental over H, ⇐⇒ H 6 H ∗ L = K for some L. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Algebraic and free extensions
r hai 6ff ha, bi 6ff ha, b, ci, and hx i 6alg hxi, ∀x ∈ F(A) ∀r ∈ Z. if r(H) > 2 and r(K ) 6 2 then H 6alg K. H 6alg K 6alg L implies H 6alg L. H 6ff K 6ff L implies H 6ff L. H 6alg L and H 6 K 6 L imply K 6alg L but not necessarily H 6alg K. H 6ff L and H 6 K 6 L imply H 6ff K but not necessarily K 6ff L.
How many algebraic extensions does a given H have in F(A) ?
Can we compute them all ? 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Takahasi’s Theorem
Theorem (Takahasi, 1951)
For every H 6fg F(A), the set of algebraic extensions, denoted AE(H), is finite.
Original proof by Takahasi was combinatorial and technical,
Modern proof, using Schreier automata, is much simpler, and due independently to Ventura (1997), Margolis-Sapir-Weil (2001) and Kapovich-Miasnikov (2002).
Additionally, AE(H) is computable. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Fringe of a subgroup
Definition Let A be a deterministic and trim A-automata, and let ∼ an eq. rel. on V A. We denote by A/ ∼ the new (deterministic and trim) A-automata resulting from identifying the vertices according to ∼, plus foldings.
Definition The fringe of A is the (finite) collection of A-automata of the form A/ ∼.
Definition
Let H 6fg F(A). The fringe of H is O(H) = {L(Γ(H)/ ∼)π |∼ eq. rel. on V A}. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Fringe of a subgroup
Observation
For H 6fg F(A), we have O(H) = {H0, H1,..., Hk }, all of them f.g., 0 0 and with H0 = H and Hk = hA i (A ⊆ A the set of used letters).
Observation
For H 6fg F(A), O(H) is finite and computable.
Proposition
For H 6fg F(A), AE(H) ⊆ O(H).
Corollary
For H 6fg F(A), AE(H) is finite. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Computing AE(H)
Corollary
For H 6fg F(A), AE(H) is computable.
1) Compute Γ(H), 2) Compute Γ(H)/ ∼ for all eq. rel. ∼ of V Γ(H), 3) Compute O(H),
4) Clean O(H) by detecting all pairs K1, K2 ∈ O(H) such that K1 6ff K2 and deleting K2. 5) The resulting set is AE(H).
For the cleaning step we need: 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Deciding free-factorness
Proposition
Given H, K 6 F(A), it is algorithmically decidable whether H 6ff K or not.
Proved by: Whitehead 1930’s (classical and exponential), Silva-Weil 2006 (faster but still exponential), Roig-Ventura-Weil 2007 (variation of Whitehead algorithm in polynomial time), Puder 2011 (graphical argument). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The algebraic closure
Observation
If H 6alg K1 and H 6alg K2 then H 6alg hK1 ∪ K2i.
Corollary
For every H 6fg K 6fg F(A), AE K (H) has a unique maximal element, called the K -algebraic closure of H, and denoted ClK (H).
Theorem Every extension H 6 K of f.g. subgroups of F(A) splits, in a unique way, in an algebraic part and a free part, H 6alg ClK (H) 6ff K. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Pseudo-varieties
Definition A pseudo-variety of groups V is a class of finite groups closed under taking subgroups, quotients and finite direct products.
i) G = all finite groups,
ii) Gp = all finite p-groups,
iii) Gnil = all finite nilpotent groups,
iv) Gsol = all finite soluble groups,
v) Gab = all finite abelian groups, k vi) for a finite group V , [V ] = all quotients of subgroups of V , k > 1. vii) ···
Definition V is extension-closed if V C W with V , W /V ∈ V imply W ∈ V. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The pro-V topology
Definition Let G be a group, and V be a pseudo-variety of finite groups. The pro-V topology on G can be defined in several equivalent ways: i) it is the smallest topology making all the morphisms from G into all V ∈ V (with the discrete topology) continuous, ii) a basis of open sets is given by ϕ−1(x), for all morphism ϕ: G → V ∈ V, iii) the normal (finite index) subgroups K E G such that G/K ∈ V form a basis of neighborhoods of 1, iv) it is the topology given by the pseudo-ultra-metric d(x, y) = 2−r(x,y), where r(x, y) = min{|V | | V ∈ V and separates x and y }.
Observation This topology is Hausdorf ⇐⇒ d is an ultra-metric ⇐⇒ G is residually-V. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The V-closure
Proposition Let G be a group equipped with the pro-V topology, and let H ≤ G. Then, TFAE: (a) H is open (b) H is clopen (i.e. open and closed)
(c) H 6fi G and G/HG ∈ V. Furthermore,
\ \ −1 clV (H) = K = ϕ (ϕ(H)). H6K , open ϕ : G→V ∈V 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The extension-closed case
Proposition (Ribes, Zaleski˘ı) Let V be an extension-closed pseudo-variety, and consider F(A) the free group on A with the pro-V topology. For a given H 6fg F(A), H is closed ⇐⇒ H is a free factor of a clopen subgroup.
Corollary
For an extension-closed V and a H 6fg F(A), we have H 6alg clV (H).
Furthermore, it can also be proven that
Proposition (Ribes, Zaleski˘ı)
In this situation, r(clV (H)) 6 r(H). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points p-closure, nil-closure
Theorem (Margolis-Sapir-Weil)
The p-closure of H 6fg F(A), clp(H), is effectively computable, for every prime p.
Theorem
For H 6fg F(A), clnil (H) = ∩pclp(H). Thus, clnil (H) is effectively computable.
Problem Find an algorithm to compute the solvable closure of a given H 6fg F(A). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Outline
1 Notation
2 Automata
3 Schreier graphs
4 First algebraic applications
5 Finite index subgroups
6 Intersections of subgroups
7 Fringe and algebraic extensions
8 The pro-V topology
9 Fixed points 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Fixed subgroups are complicated
φ: F3 → F3 a 7→ a Fix φ = ha, bab−1, cac−1i b 7→ ba c 7→ ca2
ϕ: F4 → F4 a 7→ dac b 7→ c−1a−1d −1ac Fix ϕ = hwi, where... c 7→ c−1a−1b−1ac d 7→ c−1a−1bc
w = c−1a−1bd −1c−1a−1d −1ad −1c−1b−1acdadacdcdbcda−1a−1d −1 a−1d −1c−1a−1d −1c−1b−1d −1c−1d −1c−1daabcdaccdb−1a−1. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points What is known about fixed subgroups ?
Theorem (Dyer-Scott, 75) Let φ ∈ Aut (F(A)) be a finite order automorphism of F(A). Then, Fix (φ) 6ff Fn.
Theorem (Gersten, 83 (published 87))
Let φ ∈ Aut (Fn). Then r(Fix (φ)) < ∞.
Theorem (Bestvina-Handel, 88 (published 92))
Let φ ∈ Aut (Fn). Then r(Fix (φ)) 6 n.
Theorem (Imrich-Turner, 89)
Let φ ∈ End (Fn). Then r(Fix (φ)) 6 n. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Inertia
Definition
A subgroup H 6 Fn is called inert if r(H ∩ K ) 6 r(K ) for every K 6 Fn.
Theorem (Dicks-V, 96)
Let G ⊆ Mon (Fn) be an arbitrary set of monomorphisms of Fn. Then, Fix (G) is inert; in particular, r(Fix (G)) 6 n.
Theorem (Bergman, 99)
Let G ⊆ End (Fn) be an arbitrary set of endomorphisms of Fn. Then, r(Fix (G)) 6 n.
Conjecture (V.)
Let φ ∈ End (Fn). Then Fix (φ) is inert. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points The four families
Definition
A subgroup H 6 Fn is said to be 1-auto-fixed if H = Fix (φ) for some φ ∈ Aut (Fn),
1-endo-fixed if H = Fix (φ) for some φ ∈ End (Fn),
auto-fixed if H = Fix (S) for some S ⊆ Aut (Fn),
endo-fixed if H = Fix (S) for some S ⊆ End (Fn),
Easy to see that 1-mono-fixed = 1-auto-fixed. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Relations between them
⊆ 1 − auto − fixed 6= 1 − endo − fixed
∩| ∩|
⊆ auto − fixed 6= endo − fixed
Example (Martino-V., 03; Ciobanu-Dicks, 06) −1 −1 Let F3 = ha, b, ci and H = hb, cacbab c i 6 F3. Then, H = Fix (a 7→ 1, b 7→ b, c 7→ cacbab−1c−1), but H is NOT the fixed subgroup of any set of automorphism of F3. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Relations between them
⊆ 1 − auto − fixed 6= 1 − endo − fixed
∩|k ? ∩|k ?
⊆ auto − fixed 6= endo − fixed
Conjecture (V.) 1-auto-fixed = auto-fixed, and 1-endo-fixed = endo-fixed. That is, the families of 1-auto-fixed and 1-endo-fixed subgroups are closed under intersections. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points It is true up to free factors
Theorem (Martino-V., 00)
Let S ⊆ End (Fn). Then, ∃φ ∈ hSi such that Fix (S) 6ff Fix (φ).
However... free factors of 1-endo-fixed (1-auto-fixed) subgroups need not be even endo-fixed (auto-fixed). 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Compression
Definition A subgroup H 6 F(A) is compressed when r(H) 6 r(K ) for every H 6 K 6 F(A).
Observation H inert ⇒ H compressed.
Is every compressed subgroup, inert?
Proposition
There is an algorithm which, given H 6fg F(A), decides whether H is compressed. 1. Notation 2. Automata 2. Schreier graphs 4. Algebraic appl. 5. Finite index 6. Intersections 7. Alg. ext. 6. Pro-V top. 7. Fixed points Fixed subgroups are compressed
Conjecture
There is an algorithm which, given H 6fg F(A), decides whether H is inert.
Theorem (Martino-V, 04)
Let S ⊆ End (Fn). Then, Fix (S) is compressed.
Conjecture (V.)
Let S ⊆ End (Fn). Then, Fix (S) is inert.